Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInterlacement limit of a stopped random walk trace on a torus
290
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider a simple random walk on $mathbb{Z}^d$ started at the origin and stopped on its first exit time from $(-L,L)^d cap mathbb{Z}^d$. Write $L$ in the form $L = m N$ with $m = m(N)$ and $N$ an integer going to infinity in such a way that $L^2 sim A N^d$ for some real constant $A > 0$. Our main result is that for $d ge 3$, the projection of the stopped trajectory to the $N$-torus locally converges, away from the origin, to an interlacement process at level $A d sigma_1$, where $sigma_1$ is the exit time of a Brownian motion from the unit cube $(-1,1)^d$ that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).
rate research
Read More
We study a biased random walk on the interlacement set of $mathbb{Z}^d$ for $dgeq 3$. Although the walk is always transient, we can show, in the case $d=3$, that for any value of the bias the walk has a zero limiting speed and actually moves slower than any power.
We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of modifications depends on the number of series. For the natural scaling of time and space arguments the limit process is (i) a Brownian motion if modifications are small, (ii) a linear motion with a random slope if modifications are large, and (iii) the limit process satisfies an SDE with a local time of unknown process in a drift if modifications are moderate.
We consider a random walk with a negative drift and with a jump distribution which under Cramers change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably scaled, this random walk converges in law to a nondecreasing Markov process which can be interpreted as a spectrally-positive Levy %-Khinchin process conditioned not to overshoot level one.
We consider a random walk $tilde S$ which has different increment distributions in positive and negative half-planes. In the upper half-plane the increments are mean-zero i.i.d. with finite variance. In the lower half-plane we consider two cases: increments are positive i.i.d. random variables with either a slowly varying tail or with a finite expectation. For the distributions with a slowly varying tails, we show that ${frac{1}{sqrt n} tilde S(nt)}$ has no weak limit in $De$; alternatively, the weak limit is a reflected Brownian motion.
A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk $(X_n)_{nin mathbb{N}cup{0}}$ in a sparse random environment $(S_k,lambda_k)_{kinmathbb{Z}}$ is a nearest neighbor random walk on $mathbb{Z}$ that jumps to the left or to the right with probability $1/2$ from every point of $mathbb{Z}setminus {ldots,S_{-1},S_0=0,S_1,ldots}$ and jumps to the right (left) with the random probability $lambda_{k+1}$ ($1-lambda_{k+1}$) from the point $S_k$, $kinmathbb{Z}$. Assuming that $(S_k-S_{k-1},lambda_k)_{kinmathbb{Z}}$ are independent copies of a random vector $(xi,lambda)in mathbb{N}times (0,1)$ and the mean $mathbb{E}xi$ is finite (moderate sparsity) we obtain stable limit laws for $X_n$, properly normalized and centered, as $ntoinfty$. While the case $xileq M$ a.s. for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $mathbb{E} xi=infty$ (strong sparsity) will be analyzed in a forthcoming paper.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
